Area under a Parabola by Symmetries

The area of the region under the curve over the interval equals the area of the region (in light blue) under the curve and above the line . The area-preserving shear-translation symmetry of the curve moves the region to a region whose area is one-quarter plus twice the area of the region under the curve over the interval . The scaling symmetry of the curve maps to and reduces the area by the factor . Thus so .

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Like any parabola, the parabola admits two one-parameter families of symmetries. One family consists of the scale-scale transformations , where , which scales areas by the factor . The other family consists of the "shear-translation" symmetries given by , where can be any real number; these transformations are area preserving. Applying certain subsets of these symmetries to the region , we can see that the integral in question is exactly . This approach is easily extended to determine the area of a segment of any parabola. It thus provides a calculus-free proof of the quadrature of the parabola.